(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Show that the function

f(x)

= { x/2 if x is rational

{ x if x irrational

is not differentiable at 0

2. Relevant equations

If f is differentiable at 0 then for every e > 0 there exists some d > 0 such that when |x| < d, |(f(x)-f(0))/x - L | < e for some L, which is the derivative of f at 0.

3. The attempt at a solution

Thus far I have:

Choose e = 1/4. Suppose f is differentiable at 0 and let L be f'(0). Then there is some d such that whenever |x| < d,

|(f(x)-f(0))/x - L |

= {|(x/2-0)/x - L | = | .5 - L | if x is rational

{|(x-0)/x - L | = | 1 - L | if x is irrational

Thus we have these inequalities for L:

| .5- L | < .25 and | 1 - L | < .25

which together imply that -.25 < L < .75 as well as that .75 < L < 1.25 which is a contradiction. Therefore f is not differentiable at 0.

Is this correct?

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# Homework Help: Show f(x) = { x/2 if x rational , x if x irrational is not differentiable at 0

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